Errors-in-Variables is a statistical concept that is around since decades [1]}. The usage of this concept for the learning of a neural networks was proposed before in [2]}, [3]}. In contrast to this article, those works consider a non-Bayesian framework without the quantification of uncertainties. The idea of fixing the ratio between \(\sigma _x\)  and \(\sigma _y\)  in EiV, which we propose in Section REF  below, is also quite classical [4]} and occasionally named after Deming [5]}.
Variational inference is another well-established tool from statistics that is behind much of the deep learning literature on uncertainty quantification [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. A concept building on variational inference that has some particular resemblance to this work is the one of variational autoencoders [12]}: think of \(\zeta \)  in (REF ) to be the “latent variable”. However variational autoencoders differ in their philosophy and their loss function and are therefore not directly comparable to our approach, although combining both ideas might be a way to enhance the model in (REF ). This article is structured as follows. In Section  we will discuss the generic approach presented in this work, give a proposal for the underlying priors and discuss some details useful for implementation, such as how to link \(\sigma _x\)  and \(\sigma _y\)  via a Deming factor \(\delta \)  to ensure identifiability and prevent overfitting. In Section  we discuss the results of some experiments. The focus will be laid on toy models with known ground truth to study the behavior of the EiV model in greater depth.
